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A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator. (English) Zbl 1138.81380
Summary: A nonpolynomial one-dimensional quantum potential representing an oscillator, which can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator (harmonic oscillator with a centripetal barrier), is studied. First the general case, that depends on a parameter $a$, is considered and then a particular case is studied with great detail. It is proven that it is Schrödinger solvable and then the wavefunctions ${\Psi }n$ and the energies ${E}_{n}$ of the bound states are explicitly obtained. Finally, it is proven that the solutions determine a family of orthogonal polynomials ${𝒫}_{n}\left(x\right)$ related to the Hermite polynomials and such that: (i) every ${𝒫}_{n}$ is a linear combination of three Hermite polynomials and (ii) they are orthogonal with respect to a new measure obtained by modifying the classic Hermite measure.

##### MSC:
 81Q05 Closed and approximate solutions to quantum-mechanical equations 81U15 Exactly and quasi-solvable systems (quantum theory) 33C45 Orthogonal polynomials and functions of hypergeometric type